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1.4 Lines

1.4 Lines. Essential Question: How can you use the equations of two non-vertical lines to tell whether the lines are parallel or perpendicular?. An arithmetic sequence is nothing more than a linear equation… {3, 8, 13, 18, … } → u n = 3 + (n - 1)(5) → u n = 3 + 5n – 5 → u n = 5n – 2

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1.4 Lines

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  1. 1.4 Lines Essential Question: How can you use the equations of two non-vertical lines to tell whether the lines are parallel or perpendicular?

  2. An arithmetic sequence is nothing more than a linear equation… • {3, 8, 13, 18, … }→ un = 3 + (n - 1)(5)→ un = 3 + 5n – 5→ un = 5n – 2 • If we replace n with x and un with y, we have the linear equation: y = 5x – 2 • The sequence above corresponds to the points { (1, 3), (2, 8), (3, 13), (4, 18), … } 1.4 Lines

  3. Slope • Change in y ÷ change in x • Δ y ÷Δx (delta y ÷ delta x) • Example 1 • Find the slope of the line that passes through (0,-1) and (4,1) 1.4 Lines

  4. Finding Slope From a Graph • Find two points on the coordinate plane, and use the slope formula • Properties of Slope • If m > 0, the line rises from left to right. The larger m is, the more steeply the line rises. • If m = 0, the line is horizontal • If m < 0, the line falls from left to right. The larger m is, the more steeply the line falls. 1.4 Lines

  5. Slope-Intercept Form • y = mx + b • “m” is the slope • “b” is the y-intercept (the point where the graph crosses the y-axis) • Example 3: Graphs of Arithmetic Sequences • 1st three terms of an arithmetic sequence are -2, 3, and 8. Use an explicit function and compare it to slope-intercept form 1.4 Lines

  6. Example 3 (Continued) • Sequence: -2, 3, 8, … • Explicit form: un = u1 + (n-1)(d) • u1 = , d = • What connection(s) do you see between the slope-intercept form and the original sequence? -2 5 un = -2 + (n-1)(5)un = -2 + 5n – 5un = 5n – 7 1.4 Lines

  7. Connection between Arithmetic Sequences and Lines • Explicit Form: un = u1 + (n-1)(d) • Slope Intercept Form: y = mx + b • The slope corresponds to the common difference (m = d) • The y-intercept represents the value of u0, or the term before the sequence started, which is u1 less one common difference (u1 – d) 1.4 Lines

  8. Graphing a Line • Solve an equation for y (i.e. get y by itself) • Plot the y-intercept • Use the slope (rise over run) to make a 2nd plot • Draw a line which connects the two dots (a line, not a segment) • Graphing calculator • Graph→ F1, input in the equation, solved for y • 2nd, F5 1.4 Lines

  9. Assignment • Page 40 • Problems 3 – 14 (all problems) • Show your work 1.4 Lines

  10. 1.4 Lines(Day 2) Essential Question: How can you use the equations of two non-vertical lines to tell whether the lines are parallel or perpendicular?

  11. Point-Slope Form • Use point slope form when you’re given a point and a slope (SURPRISE!) or two points to determine an equation (and use those points to determine the slope) • y – y1 = m(x – x1) • Any point given can be used for (x1, y1) 1.4 Lines

  12. Point-Slope Form (Example 6) • Find the equation of the line that passes through the point (1, -6) with a slope 2. y – y1 = m(x – x1) y – (-6) = 2(x – 1) y + 6 = 2x – 2 – 6 – 6 y = 2x – 8 1.4 Lines

  13. Vertical and Horizontal Lines • Equation of a Horizontal Line • Every point along a horizontal line will have the same y value. • Written as y = b (where b is the y-intercept) • The slope of a horizontal line = 0 • Equation of a Vertical Line • Every point along a vertical line will have the same x value. • Written as x = c (where c is some constant) • The slope of a vertical line is undefined 1.4 Lines

  14. Parallel & Perpendicular Lines • Parallel lines have the same slope • Perpendicular lines have inverse reciprocal slopes • That means the product of the slopes of two perpendicular lines is -1 • Take the slope of one line, flip as a fraction and flip sign. 1.4 Lines

  15. Parallel & Perpendicular Lines (Example 9) • Given line M whose equation is 3x - 2y + 6=0, find the equation of the parallel and perpendicular lines which go through the point (2, -1) • Parallel Line • Get y by itself to findthe slope of line M • The slope of M is 3/2 • Use point-slope form y – y1 = m(x – x1) 3x – 2y + 6 = 0 y – (-1) = 3/2(x – 2) –3x –6 –3x – 6 -2y = -3x – 6 y + 1 = 3/2x – 3 -2 -2 -2 – 1 – 1 y = 3/2x + 3 y = 3/2x – 4

  16. Parallel & Perpendicular Lines (Example 9) • Given line M whose equation is 3x - 2y + 6=0, find the equation of the parallel and perpendicular lines which go through the point (2, -1) • Perpendicular Line • Slope of M = 3/2, so the perpendicular slope is -2/3 • Use point-slope form y – y1 = m(x – x1) y – (-1) = -2/3(x – 2) y + 1 = -2/3x + 4/3 – 1 – 1 y = -2/3x + 1/3

  17. Standard Form of a Line • Written as Ax + By = C, where A, B, and C are integers • We can convert our perpendicular line to standard form by removing any fractions 1.4 Lines

  18. Forms of Linear Equations • Standard Form: Ax + By = C • All integers, x and y terms on same side • Slope-Intercept Form: y = mx + b • Best for quickly identifying slope and graphing • Point-Slope Form: y – y1 = m(x – x1) • Best for writing equations • Horizontal Lines: y = b • Slope = 0 • Vertical Lines: x = c • Slope is undefined 1.4 Lines

  19. Assignment • Page 40 – 41 • Problems 17 – 49 (odd problems) • Show your work 1.4 Lines

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